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Moore graphs and beyond: A survey of the degree/diameter problem
 ELECTRONIC JOURNAL OF COMBINATORICS
, 2013
"... The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bo ..."
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Cited by 26 (4 self)
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The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem ‘from above’, remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem ‘from below’. This survey aims to give an overview of the current stateoftheart of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moorelike bounds for special types of graphs and digraphs, such as vertextransitive, Cayley, planar, bipartite, and many others, on
Large Cayley Graphs and Digraphs with Small Degree and Diameter
, 1995
"... We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter. ..."
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Cited by 11 (0 self)
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We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter.
Digraphs of degree 3 and order close to the Moore bound
, 1995
"... It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist (see [20] or [5]). Furthermore, for degree 2, it is shown that for k 3 there are no digraphs of order `close' to, i.e., one less than, Moore bound [18]. In this paper, we shall consider digraphs of diameter k, degree 3 ..."
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Cited by 9 (6 self)
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It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist (see [20] or [5]). Furthermore, for degree 2, it is shown that for k 3 there are no digraphs of order `close' to, i.e., one less than, Moore bound [18]. In this paper, we shall consider digraphs of diameter k, degree 3 and number of vertices one less than Moore bound. We give a necessary condition for the existence of such digraphs and, using this condition, we deduce that such digraphs do not exist for infinitely many values of the diameter. Keywords  digraphs, Moore bound, diameter, degree. 1. Introduction By a digraph we mean a structure G = (V; A) where V (G) is a nonempty set of distinct elements called vertices; and A(G) is a set of ordered pairs (u; v) of distinct vertices u; v 2 V called arcs. The order of a digraph G is the number of vertices in G, i.e., jV (G)j. An inneighbour of a vertex v in a digraph G is a vertex u such that (u; v) 2 G. Similarly, an outneighbour of a vertex v is a v...
On the structure of digraphs with order close to the Moore bound
, 1996
"... The Moore bound for a diregular digraph of degree d and diameter k is M d;k = 1 + d + : : : + d k . It is known that digraphs of order M d;k do not exist for d ? 1 and k ? 1 ([24] or [6]). In this paper we study digraphs of degree d, diameter k and order M d;k \Gamma 1, denoted by (d; k)digraphs ..."
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Cited by 7 (5 self)
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The Moore bound for a diregular digraph of degree d and diameter k is M d;k = 1 + d + : : : + d k . It is known that digraphs of order M d;k do not exist for d ? 1 and k ? 1 ([24] or [6]). In this paper we study digraphs of degree d, diameter k and order M d;k \Gamma 1, denoted by (d; k)digraphs. Miller and Fris showed that (2; k) digraphs do not exist for k 3 [22]. Subsequently, we gave a necessary condition of the existence of (3; k)digraphs, namely, (3; k)digraphs do not exist if k is odd or if k + 1 does not divide 9 2 (3 k \Gamma 1) [3]. The (d; 2)digraphs were considered in [4]. In this paper, we present further necessary conditions for the existence of (d; k)digraphs. In particular, for d; k 3, we show that a (d; k)digraph contains either no cycle of length k or exactly one cycle of length k. 1 Introduction By a digraph we mean a structure G = (V; A) where V (G) is a nonempty set of elements called vertices; and A(G) is a set of ordered pairs (u; v) of disti...
Large graphs with small degree and diameter: A voltage assignment approach
"... Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. The main objective of the paper is to revisit the classical degree/diameter problem for graphs from this new perspective. We derive a fairly general up ..."
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Cited by 5 (2 self)
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Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. The main objective of the paper is to revisit the classical degree/diameter problem for graphs from this new perspective. We derive a fairly general upper bound on the diameter of a lift in terms of the properties of the base voltage graph, and prove some results on vertextransitive lifts. The potential of the new method is highlighted by showing that all currently known largest Cayley graphs (of given degree and diameter) for semidirect products of cyclic groups can be described by means of a voltage assignment construction, using simpler groups. This research started when J. Plesn'ik and J. Sir'an were visiting the Department of Computer Science of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. 1 Introduction In the past few decades there has been growing interest in the design of interconnection ...
Synthesis of Interconnection Networks: A Novel Approach
 In Proc. of the 20th International Conference on Dependable Systems and Networks
, 2000
"... The interconnection network is a crucial element in parallel and distributed systems. Synthesizing networks that satisfy a set of desired properties, such as high reliability, low diameter and good scalability is a dicult problem to which there has been no completely satisfactory solution. ..."
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Cited by 4 (2 self)
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The interconnection network is a crucial element in parallel and distributed systems. Synthesizing networks that satisfy a set of desired properties, such as high reliability, low diameter and good scalability is a dicult problem to which there has been no completely satisfactory solution.
A note on constructing large Cayley graphs of given degree and diameter by voltage assignments
, 1997
"... Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diame ..."
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Cited by 4 (1 self)
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Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree # 15 and diameter # 10 have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. # This research started when J. Plesnk and J. Siran were visiting the Department of Computer Science and Software Engineering of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. the electronic journal of combinatorics 5 (1998), #R9 2 This opens up a new possible direction in the search for large vertextransitive graphs of given degree and diameter....
Matrix Techniques for Strongly Regular Graphs and Related Geometries
"... this paper gives a survey of the recent results on strongly regular graphs. It is a sequel to Hubaut [27] earlier survey of constructions. Seidel [35] gives a good treatment of the theory. 2 Association schemes ..."
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Cited by 3 (0 self)
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this paper gives a survey of the recent results on strongly regular graphs. It is a sequel to Hubaut [27] earlier survey of constructions. Seidel [35] gives a good treatment of the theory. 2 Association schemes
Moore Graphs, Latin Squares and Outer Automorphisms
, 2003
"... A connection between Moore graphs, outer automorphisms of symmetric groups, and families of almost orthogonal latin squares is exhibited. ..."
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Cited by 2 (0 self)
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A connection between Moore graphs, outer automorphisms of symmetric groups, and families of almost orthogonal latin squares is exhibited.
Dynamic Cage Survey
, 2008
"... A (k, g)cage is a kregular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant const ..."
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Cited by 2 (0 self)
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A (k, g)cage is a kregular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.